Since there seems to be no progress in this interesting question, I took the liberty to reformulate it in a way, that is easier to understand. Moreover, my answer shows that the question is related to amenable groups. Therefore I changed the title to make it more attractive to group theorists.

Let $C_0(\mathbb R)=$ { $ f:\mathbb R\rightarrow \mathbb R $ } is the subset of $C(\mathbb R)$ consisting of functions such that for every $ε > 0$, there is a compact set $K⊂\mathbb R$ such that $|f(x)| < ε$ for all $x \in \mathbb R\setminus K $. This is usually called the space of functions vanishing at infinity.

There are two homeomorphisms of the line $g_1, g_2$ and a continuous linear positive functional $l$ on $C_b(\mathbb R)$ which is invariant with respect to $g_1, g_2$. Also this functional is permanent: $l([C_0(\mathbb R)])=0$, so $l$ is "concentrate at infinity".

Then we make a Stone-Čech compactification of the $\mathbb R $ designated as $\beta \mathbb R $.

After Stone-Čech compactification of the line, the homeomorphism still will be a homeomorphism and I can show that it will transfer $\mathbb R$ to $\mathbb R$ and the remainder $\mathbb R^* $ to $\mathbb R^* $ ($\mathbb R^* = \beta\mathbb R\setminus\mathbb R $). By the Riesz representation theorem, for our linear functional (already on $\beta\mathbb R$ and still invariant) there is a unique regular countably additive Borel measure $\mu$ on $\beta\mathbb R$. I can show that this measure will be trivial zero at $\mathbb R$. I need to understand under which conditions on the homeomorphisms this measure will be trivial zero at $\mathbb R^* $. I will be very grateful for links on this problem.

UPDATE [12.06.2012]
There is a potential result: let $g_1, g_2\in Homeo_+(\mathbb R)$. $g_1$ can be represented as a line $y=x+k$, where $k>0$, and $g_2$ is such that there are two points $t_1,t_2$, for which following conditions are fulfilled $ t_1 < t_2; g_2(t_1)=t_1; g_2(t_2)=t_2; g_2(t)>t$ (or $ g_2(t) < t $ ) for $t \in (t_1, t_2); g_1(t_1) \in (t_1, t_2); g_2(t)$ is an arbitrary monotone increasing curve for $ t\in (-\infty, t_1) \cup (t_2,+\infty) $. Also we have a group $G =< g_1,g_2 >$ with two generators.

If $L$ is a continuous linear functional, which is invariant under $g_1, g_2$, and $L$ was "lowered" from the group $G$ to the $\mathbb R$, and after "lowering" it appears to be permanent (the restriction of $L$ to $C_0(\mathbb R)$ is zero), then $L=0$ and the group $G$ is not amenable.

The definition of "lowering" functional from the group to $\mathbb R$ can be found in section §3.1 of the paper http://arxiv.org/abs/1112.1942 and the proof of the statement is the whole paper. This paper has not yet verified.

Hi Mariarty. I for one don't understand the question. What "initial problem" are you referring to? I guess it's another question you asked here, in which case it would be helpful to provide a link. Also, it would help to define some of your notation (such as C_b), and to split your question into paragraphs.
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Tom LeinsterMar 5 '12 at 13:22

PS - in case you hadn't noticed, there's an "edit" button with which you can do all this.
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Tom LeinsterMar 5 '12 at 13:22

@Tom Leinster I think that subject of the "initial problem" doesn't matter. I could not find any references related to measures on remainder and I described the whole problem without expecting help in solving but in hope that someone met something like this in literature.
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MariartyMar 5 '12 at 15:27

To my understanding, if $\mu$ is zero on $\mathbb{R}$ and on $\beta \mathbb{R} \setminus \mathbb{R}$ then $\mu=0$ and hence $l=0$. Conversely, if $l \in C_b(\mathbb{R})^\ast = C(\beta\mathbb{R})^\ast$ is zero, then the corresponding Radon measure is also zero. So, if I'm not missing something, you are looking for conditions on $g_i$ that imply $l=0$ ?
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RalphMar 5 '12 at 15:58

In order to obtain $L_0$ let $C_l(\mathbb{R})$ be the space of the continuous
functions $f: \mathbb{R} \to \mathbb{R}$ such that
$\lim_{t \to +\infty}f(t)$ exists. Then
$$L_0: C_l(\mathbb{R}) \to \mathbb{R},\; f \mapsto \lim_{t \to +\infty}f(t)$$
is continous, linear with $L_0|C_0(\mathbb{R}) = 0$ and $L_0(1)=1$.
Finally, since $C_l(\mathbb{R})$ is a closed subspace of $C_b(\mathbb{R})$, $L_0$
can be continuously extended to $C_b(\mathbb{R})$ by the Hahn-Banach theorem.

@Ralph Great thanks for realized calculations. But your result is exactly what I called as "initial problem". I've already done the way back from your result to my question. But this is good news: I'm on the right path. Thank you again, question can be closed.
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MariartyMar 22 '12 at 18:47

Good to hear that you are on the right path. Would you mind sharing your insights with us, maybe in form of an arxiv-link or so, when you finished your work on the problem ?
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RalphMar 22 '12 at 19:18